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I was asked, What is the difference between putting the negation outside of parenthesis and putting into individual brackets?

Let's have a look.

‘What is the difference between putting the negation outside of parenthesis and putting into individual brackets?’

Consider this sentence, 'Either the music has stopped or I am dead.'

Either the music has stopped or I am dead.

A ∨ B

Let's suppose we can represent it in awFOL as A ∨ B.

What happens if you deny the whole sentence?

That’s not true.

To capture this denial in awFOL, we put negation outside the brackets.

¬(A ∨ B)Now consider this alternative sentence, 'Either the music has not stopped or I am not dead.'

Either the music has not stopped or I am not dead.

To capture this in awFOL, we put negation inside the brackets.

¬A ∨ ¬BSo now we have a sentence with negation outside the brackets and another one with negation inside the brackets.

And, thinking about the English sentences, you can perhaps see that if we put negation inside the brackets we are not denying the sentence, just asserting a different disjunction.

Let's see how these two differ by constructing truth-tables.

In constructing truth-tables we start with the truth table for disjunction (∨).

A | B | A ∨ B | ¬(A ∨ B) | ¬A | ¬B | ¬A ∨ ¬B |

T | T | T | F | F | F | F |

T | F | T | F | F | T | T |

F | T | T | F | T | F | T |

F | F | F | T | T | T | T |

To get the truth table for ¬(A ∨ B) we merely need to flip the truth-values of the truth-table for (A ∨ B).

By contrast, if we want the truth-table for ¬A ∨ ¬B, we need to start with truth-tables for ¬A and for ¬B.

We then combine these two truth tables using disjunction -- so it's True whenever ¬A is True or ¬B is True, and False otherwise.

Now focus just on the truth tables for our two sentences, one with negation inside the brackets the other with negation outside the brackets

The truth of the first denial sentence, ¬(A ∨ B), guarantees that you're alive (B = 'I am dead'), whereas the truth of the second denial, ¬A ∨ ¬B, doesn't entail that you're alive.

Sometimes students decide they'll ignore brackets and hope for the best. But the difference between putting the negation outside of parenthesis and putting into individual brackets is a matter of life or death.

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5.18